Consider a polygon lying in the Euclidean plane with labeled edge lengths. The moduli space of polygons is the space of all polygons with the same labeled edge lengths, modulo orientation preserving isometries. It is well known that this space is generically a smooth manifold. For certain combinations of edge lengths, however, singularities can arise. We show these singularities to be isolated, with a neighborhood of each singularity homeomorphic to a cone over a product of spheres. We proceed to explicitly compute the singular topologies that arise as moduli spaces of pentagons. We then turn our attention to the number of different (up to diffeomorphism) smooth topologies that can arise as moduli spaces of polygons. It is known that, for a fixed number of edges, the number of possible topologies is finite. The exact number is only known up to the case of pentagons, however. We provide a new structure that summarizes the possible topologies of a moduli space of polygons in a directed graph. We then use this structure to provide bounds on the number of topologies that can arise as moduli spaces of polygons with no more than eight edges.